Question

What would be the capital allocation between the risk free asset and the optimal risky investment portfolio for an individual with risk aversion coefficient of 3 ?

If the initial investment is $100,000, how much money should the investor allocate to each of the 5 assets (risk free asset and 4 risky assets). RF RATE 2% YEARLY

Expected Avarage Return			optimal investment portfolio in the risky			global minimum variance portfolio.
			Weight			Weight
PG	0.010848		PG	0.2053		PG	0.336
Microsoft	0.014854		Microsoft	0.223		Microsoft	0.0065
BAC	0.011589		BAC	0.5203		BAC	0.632
Exxon	0.012043		Exxon	0.0514		Exxon	0.0255
				1.000000			1.000000
Variance
			E[r]	0.01218804		E[r]	0.011372468
PG	0.004478		Portfolio Variance	0.003034717		Portfolio Variance	0.003082286
Microsoft	0.012820		Std Dev	0.055088262		Std Dev	0.055518341
BAC	0.005611		Sharp	0.190991196		Sharp	0.174821525
Exxon	0.002820
Covariance
Cov(PG, Microsoft)	-0.000649
Cov(PG, BAC)	0.000683
Cov(PG, Exxon)	0.000433
Cov(Microsoft, BAC)	0.001681
Cov(Microsoft, Exxon)	0.000804
Cov(BAC, Exxon)	0.000757

My question is . how does this guy get .00781. From the table above.?

To maximize utility,

Capital allocation between risky and risk free asset are calculated as below.

Weight of risk asset=E(R_Rp) +E(Rf)/3*(.055088262)²

^{=-.00781/.00910}

^=-.85824

^{-85.824% invested in risky asset and 185.824%in risk free asset.}

Answer 1

optimal portfolio consists of risky assets portfolio and risk free asset. Let suppose "w1" is the weight of risky asset portfolio, so remaining weight in optimal porfolio would be of risk free asset i.e. (1-w1)

optimal portfolio return = w1* risky asset + (1-w1)*risk free asset

To simplify the equation, we get

optimal portfolio return - risk free asset = w1(risky asset - risk free asset) -------------------------- (1)

optimal portfolio return is given as 0.01218804

risk free asset return is given as 2% = 0.02

Your question is how they got 0.00781

if you enter the above value in left hand side of equation (1), you get

0.01218804 - 0.02 = -0.0078116.

I hope you have already understood the other part of the given solution you have provided in the above question.

Thank you.